3.816 \(\int \frac{x^3 \sqrt{1+x}}{(1-x)^{5/2}} \, dx\)

Optimal. Leaf size=78 \[ \frac{2 \sqrt{x+1} x^3}{3 (1-x)^{3/2}}-\frac{13 \sqrt{x+1} x^2}{3 \sqrt{1-x}}-\frac{1}{6} \sqrt{1-x} \sqrt{x+1} (33 x+52)+\frac{11}{2} \sin ^{-1}(x) \]

[Out]

(-13*x^2*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*x^3*Sqrt[1 + x])/(3*(1 - x)^(3/2)) -
(Sqrt[1 - x]*Sqrt[1 + x]*(52 + 33*x))/6 + (11*ArcSin[x])/2

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Rubi [A]  time = 0.121988, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{x+1} x^3}{3 (1-x)^{3/2}}-\frac{13 \sqrt{x+1} x^2}{3 \sqrt{1-x}}-\frac{1}{6} \sqrt{1-x} \sqrt{x+1} (33 x+52)+\frac{11}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(x^3*Sqrt[1 + x])/(1 - x)^(5/2),x]

[Out]

(-13*x^2*Sqrt[1 + x])/(3*Sqrt[1 - x]) + (2*x^3*Sqrt[1 + x])/(3*(1 - x)^(3/2)) -
(Sqrt[1 - x]*Sqrt[1 + x]*(52 + 33*x))/6 + (11*ArcSin[x])/2

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Rubi in Sympy [A]  time = 10.824, size = 68, normalized size = 0.87 \[ \frac{2 x^{3} \sqrt{x + 1}}{3 \left (- x + 1\right )^{\frac{3}{2}}} - \frac{13 x^{2} \sqrt{x + 1}}{3 \sqrt{- x + 1}} - \frac{\sqrt{- x + 1} \sqrt{x + 1} \left (\frac{33 x}{2} + 26\right )}{3} + \frac{11 \operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(1+x)**(1/2)/(1-x)**(5/2),x)

[Out]

2*x**3*sqrt(x + 1)/(3*(-x + 1)**(3/2)) - 13*x**2*sqrt(x + 1)/(3*sqrt(-x + 1)) -
sqrt(-x + 1)*sqrt(x + 1)*(33*x/2 + 26)/3 + 11*asin(x)/2

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Mathematica [A]  time = 0.0693957, size = 52, normalized size = 0.67 \[ 11 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{\sqrt{1-x^2} \left (3 x^3+12 x^2-71 x+52\right )}{6 (x-1)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*Sqrt[1 + x])/(1 - x)^(5/2),x]

[Out]

-(Sqrt[1 - x^2]*(52 - 71*x + 12*x^2 + 3*x^3))/(6*(-1 + x)^2) + 11*ArcSin[Sqrt[1
+ x]/Sqrt[2]]

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Maple [A]  time = 0.024, size = 97, normalized size = 1.2 \[{\frac{1}{6\, \left ( -1+x \right ) ^{2}} \left ( -3\,{x}^{3}\sqrt{-{x}^{2}+1}+33\,\arcsin \left ( x \right ){x}^{2}-12\,{x}^{2}\sqrt{-{x}^{2}+1}-66\,\arcsin \left ( x \right ) x+71\,x\sqrt{-{x}^{2}+1}+33\,\arcsin \left ( x \right ) -52\,\sqrt{-{x}^{2}+1} \right ) \sqrt{1-x}\sqrt{1+x}{\frac{1}{\sqrt{-{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(1+x)^(1/2)/(1-x)^(5/2),x)

[Out]

1/6*(-3*x^3*(-x^2+1)^(1/2)+33*arcsin(x)*x^2-12*x^2*(-x^2+1)^(1/2)-66*arcsin(x)*x
+71*x*(-x^2+1)^(1/2)+33*arcsin(x)-52*(-x^2+1)^(1/2))*(1-x)^(1/2)*(1+x)^(1/2)/(-1
+x)^2/(-x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*x^3/(-x + 1)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.241432, size = 273, normalized size = 3.5 \[ -\frac{3 \, x^{7} + 3 \, x^{6} - 79 \, x^{5} + 285 \, x^{4} - 88 \, x^{3} - 396 \, x^{2} +{\left (3 \, x^{6} + 24 \, x^{5} - 87 \, x^{4} - 44 \, x^{3} + 396 \, x^{2} - 264 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 66 \,{\left (x^{5} + 2 \, x^{4} - 11 \, x^{3} + 4 \, x^{2} -{\left (x^{4} - 5 \, x^{3} + 12 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x - 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 264 \, x}{6 \,{\left (x^{5} + 2 \, x^{4} - 11 \, x^{3} + 4 \, x^{2} -{\left (x^{4} - 5 \, x^{3} + 12 \, x - 8\right )} \sqrt{x + 1} \sqrt{-x + 1} + 12 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*x^3/(-x + 1)^(5/2),x, algorithm="fricas")

[Out]

-1/6*(3*x^7 + 3*x^6 - 79*x^5 + 285*x^4 - 88*x^3 - 396*x^2 + (3*x^6 + 24*x^5 - 87
*x^4 - 44*x^3 + 396*x^2 - 264*x)*sqrt(x + 1)*sqrt(-x + 1) + 66*(x^5 + 2*x^4 - 11
*x^3 + 4*x^2 - (x^4 - 5*x^3 + 12*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 12*x - 8)*arc
tan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 264*x)/(x^5 + 2*x^4 - 11*x^3 + 4*x^2 - (
x^4 - 5*x^3 + 12*x - 8)*sqrt(x + 1)*sqrt(-x + 1) + 12*x - 8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(1+x)**(1/2)/(1-x)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22474, size = 66, normalized size = 0.85 \[ -\frac{{\left ({\left (3 \,{\left (x + 2\right )}{\left (x + 1\right )} - 86\right )}{\left (x + 1\right )} + 132\right )} \sqrt{x + 1} \sqrt{-x + 1}}{6 \,{\left (x - 1\right )}^{2}} + 11 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)*x^3/(-x + 1)^(5/2),x, algorithm="giac")

[Out]

-1/6*((3*(x + 2)*(x + 1) - 86)*(x + 1) + 132)*sqrt(x + 1)*sqrt(-x + 1)/(x - 1)^2
+ 11*arcsin(1/2*sqrt(2)*sqrt(x + 1))